EXCHANGE 


UNIVERSITY  OF  PENNSYL\'AXIA 


THE   RESOLVENTS  OF  KONIG 
AND  OTHER  TYPES  OF  SYM- 
METRIC FUNCTIONS 


BY 

STANLEY  P.  SHUGERT 


A  THESIS 

PRESENTED  ;rO  THE  FACULTY  OF  THE  GRADUATE  SCHOOL 

IN  PARTIAL  FULFILMENT  OF  THE  REQUIREMENTS  FOR 

THE  DEGREE  OF  DOCTOR  OF  PHILOSOPHY 


PRESS  OP 

THE  NEW  ERA  PRINTING  COMPANY 

LANCASTER,  PA. 


UNIVERSITY  OF  PENNSYLVANIA 


THE   RESOLVENTS  OF  KONIG 
AND  OTHER  TYPES  OF  SYM- 
METRIC FUNCTIONS 


BY 

STANLEY  P.  SHUGERT 


A  THESIS 

PRESENTED   TO   THE  FACULTY   OF   THE  GRADUATE  SCHOOL 

IN  PARTIAL  FULFILMENT  OF  THE  REQUIREMENTS  FOR 

THE   DEGREE   OF  DOCTOR  OF  PHILOSOPHY 


PRESS  OP 

THE  NEW  ERA  PRINTING  COMPANY 

LANCASTER,  PA. 


The  author  wishes  to  acknowledge  his  indebtedness  to 
Professor  0.  E.  Glenn  at  whose  suggestion  this  paper  was 
undertaken  for  his  helpful  supervision  and  encouragement. 


44432G 


THE  RESOLVENTS  OF  KONIG  AND  OTHER 
TYPES  OF  SYMMETRIC  FUNCTIONS. 

By  Stanley  P.  Shugert. 


INTRODUCTION. 

Symmetric  functions  of  the  differences  of  the  roots  of  an  equa- 
tion are  an  important  part  of  many  theories,  particularly  of 
binary  invariant  theory,  and  much  attention  has  been  given  to 
the  computation  of  such  functions;  while  the  functions  which 
are  not  symmetric  in  differences  have  received  very  little  atten- 
tion, but  in  some  instances  have  an  important  bearing  in  general 
theories.  As  a  particular  instance,  there  are  the  resolvents  of 
degree  ('")  of  an  equation, 

fix)  =  a;*"  +  Aix"^-^  +  AiX^-^  + \-  A„,  =  0, 

which  were  studied  by  J.  Konig  in  the  Mathematische  Annalen, 
Vol.  15  (1879).  In  brief,  these  are  constructed  as  follows:  the 
roots  of 

F(z)  =  f(z  -\-f,)  =  z^  +  B,z^-'  -{-■-■-{■  Bm=0 

are  (ai  —  fx)  {i  =  1,  2,  •  •  •  m},  where  the  ai  are  the  roots  of 
f{x)  =  0.  The  equation  whose  roots  are  the  ("')  quantities  of 
the  type, 

(-  ma,  -  /x)(a2  -  m)  •  •  •  («r  -  m)     (r  =  1,  2,  . . .  m), 
is  rational  in  Bi  and  is  a  resolvent  of  f{x)  =  0. 

In  volume  15  (1914)  of  the  Transactions  of  the  American  Mathe- 
matical Society,  Prof.  O.  E.  Glenn  has  published  a  paper  in  which 
is  given  the  expansion  of  the  binary  form  of  order  m  =  fxv, 

fix)  =  aox{^  +  aia-r-^a  +  •  •  •  +  amX2"' 

as  a  power  series  in  which  the  argument  is  the  binary  form  of 
order  v, 

^x"  =  ^oa^i"  +  ^iXi'-'Xi  4-  .  . .  +  ^^a:2^ 

In  §  2  of  this  paper,  we  obtain  an  independent  proof  of  his  result 

1 


THE  RESOLVENTS  OP  KONIG. 


for  J'  =  2  by  a  method  depending  upon  the  solution  of  a  system 
of  linear  equations  and  a  preliminary  lemma  on  determinants, 
§  1.  The  remainder  of  the  present  paper  is  devoted  to  a  succes- 
sion of  applications,  introducing  theories,  which  depend  upon 
this  expansion  for  the  case  v  =  2,  and  finally  introducing  in  §  6 
a  method  for  the  determination  of  the  Konig  Resolvents  with 
the  aid  of  the  tables  given  in  §  7.  In  §  3  a  minimum  set  of 
seminvariant  conditions  that  a  given  quadratic  form  may  divide 
a  given  form  /  is  obtained  and  §  4  is  devoted  to  a  problem  in 
rational  fractions.  There  are  obtained  in  §  5,  methods  for  the 
computation  of  symmetric  functions  of  the  sums  and  of  the 
products  of  the  roots  of  an  equation,  taken  in  pairs. 

§1.    A  LEMMA  ON  DETERMINANTS. 
We  first  prove  the  following  lemma  concerning  determinants: 
Lemma:    The  determinant  {ckrs]  in  which  every  constitiient  is 
zero  for  s  >  r  -\-  1  and  not  zero  fors=r  can  be  reduced  to  a  deter- 
minant {Ofg}  in  which  every  constituent  is  zero  for  s  >  r. 


Let 


Ors 


(A) 


an, 

^21, 


ai2, 

^22, 


0 

023, 


Oil,      ai 


0, 

0 

••    0 

0, 

0 

••    0 

(ti  i+1. 

0 

..  0 

••  0 

Or-l  i+1, 

•  •    flr-l  r 

Or  t+l, 

••    Orr 

Or-i  1,    Or-i  i,    Or-i  3, 
Or  1  Or  2,         Or  3, 

Multiply  the  last  row  by  Or-i  rjor  r  and  subtract  from  the  pre- 
ceding row.    This  will  give 


(B) 


On, 
O21, 


0.1, 


021  0, 

022  O23, 

ail 


0,      0 
0,      0 


(ar-l(ar)r)l    (Or_i(ar)r)2  (ar_i(ar)r)i+l  (Or-l(Or)r)r- 


0 
0 
0 
0 
0 

-,  0 


arr  arr  arr  Orr 

Orl,  Or2,  •••  (h  i+1,         ••'  Orr-l,  ttrr 

in  which  (ar_i(ar)r)f+i  is  a  symbolic  notation  for  the  second  order 


A  LEMMA  ON  DETERMINANTS.  3 

determinant  (ar-i  i+iO>rr  —  dr-i  rdr  i+i).  The  minor  determinant 
Arr  is  of  the  same  type  as  the  determinant  A  and  by  applying 
the  same  process  to  it  and  continuing  the  method  for  each 
succeeding  minor  determinant,  we  finally  obtain: 


(aiiazitts  '  •  •  {ar)r  •  •  •)3)2)l, 


0,  •  •  •  0 


azias  '  '  •  {ar)r  ■  • •)3)2       * 
{aiiasia^  •  •  •  {ar)r  •  •  •)4)3)i        Mas  •  •  •  (ar)r  •  •  Os) 


(a3(«4  •••    («r)r   •••)4)3       '  («3(«4  '  *  *    Wr  *  *  '  )4)3  * 

ai{ai+i  •  '  '  {ar)r  •  •  ')i+l)l  (at(«t+i  •  •  •  Mr  -  •  •)i+l)2 


{ai+i{ai+2  •  •  •    Mr  •  •  •)i+2)i+l'    ai+i(ai+2   •  •  •    Mr  '  •  ')i+2)i+l' 

ai{ai+i  •  ••  Mr  • '  ')i+i)j  Q 

(ai+l(«i+2   •  •  •    i0,r)r  '  "  •)i+2)i+l' 


(ar-iMr)l  ar-lMr)2 

> 


"0 


arr  Orr 

Orl,  Ort,  "  '  drr 

(^  =  1,  2,  .  • .  r,  i  =  1,  2,  . . .  i). 

The  interlacing  determinant  notation  is  merely  an  extension 
of  the  symbolic  method  explained  above  for  a  determinant  of  the 
second  order.  These  determinants  are  expanded  by  means  of  a 
succession  of  second  order  determinants  and  the  method  as 
shown  for  (ar-i(ar)r)i+i  =  (cfr-i  i+i«rr  —  ar_,>ari+i)  is  to  take  the 
determinant  subscripts,  those  written  outside,  first  in  an  order 
symmetric  with  respect  to  the  center  with  the  product  of  the 
constituents  and  then  to  subtract  the  product  obtained  by  re- 
versing the  determinant  subscripts.  Except  in  the  last  reduction, 
two  constituents  of  the  second  order  determinants  will  be  sym- 
bolic determinants. 

Corollary:   The  determinant  {ar8}s>r+i  can  be  written  as  a 
symbolic  determinant:  i.  e. 

{«rsls>r+i  =  {ai{a2{az{ai  •  •  •  Mr  ' '  04)3)2)1. 


4  THE   RESOLVENTS   OF   KONIG. 

§2.    EXPANSION  OF  A  POLYNOMIAL  AS  A  POWER 
SERIES  OF  A  QUADRATIC  POLYNOMIAL. 

Theorem:  Any  form  of  degree  m  can  be  expressed  in  terms  of  a 
power  series  of  a  quadratic  form  with  linear  expressions  for  the 
coefficients. 

First:  if  m  is  even,  m  =  2n,  we  may  write, 

a^"*  =  aox"^""  +  aia:2n-i  _|_  a^a^""-^  + h  «2n  =  («i  +  cax) 

(1)  +  (as  +  a,x)^.'  +  •  •  •  +  (a2r-i  +  a^rXm^r-'^ 

where  ^^^  =  ^o^^^  +  ^ix  +  ^2- 

Since  ao  =  ^o"*  and  is  determined,  the  number  of  a's  on  the 
right  is  equal  to  the  number  of  a's  on  the  left  and  since  the  coeffi- 
cients of  the  powers  of  x  on  the  right  are  linear  in  the  a's,  we  have 
upon  equating  coefficients  2n  =  m  linear  non-homogeneous  equa- 
tions in  the  2n  quantities  ai. 

For  the  case  2n  =  4,  we  have 

iaQaia2azai){xy  =  (ai  +  oi2x)  +  (as  +  a4x){^QX^  +  ^la;  +  ^2) 

and  upon  multiplying  out  and  rearranging, 

^oai  =  ai  —  2^o^i, 
^otts  +  ^10:4  =  02  —  ^1^  —  2^o^2, 
«2  +  ?i«3  +  ^2014  =  as  —  2^i^2, 

ai  +  ^2^3  =   04  —   ^2^. 

Second:  If  m  is  odd  m  =  2n  —  1,  we  can  write 

ax^  =  «oa:"^~^  +  Oia^^^-^  +  02^:2'*-^  H h  a2n-i 

(la)  =  (ai  4-  Q;2a^)  +  (aa  +  aiX)^J^  +  •  •  • 

+  (a2r-l  +  a2rX)aJ'y-'  +    '  '  '  +  («2n-l  +  A;^:)  (^,2)n-l^ 

where  A;  is  determined  to  be  ao/^o"~^  and  we  have  upon  equating 


A   QUADRATIC   POLYNOMIAL.  5 

coefficients  2n  —  1  linear  non-homogeneous  equations  to  de- 
termine the  2n  —  1  a's. 
For  the  case  m  =  3,  we  have 

(aoaia2az){xy  =  (ai  +  aix)  +  {as  +  kx){^oX^  +  ^ix  +  ^2) 

and  upon  multiplying  out  and  rearranging, 

^0«3  =   «!  —   ^b 
«2  +   ^l^S  =    02  —   ^2, 

«i  +  ^20:3  =  as, 

if,  as  before,  we  take  ao  =  ^o'*~S  which  we  may  do  without  loss 
of  generality. 

The  particular  form  that  these  equations  take,  enable  us  to 
express  them  in  general  in  terms  of  an  operator,  when  we  observe 
that  any  power  of  a  form  is  a  covariant  of  that  form  and  so  is 
given  by  its  end  term  and  a  partial  derivative  operator.  Hence 
the  equations : 

2)o^^[(».-i)/2]^  =  fli  -  D^f^"'"', 

. . .  -f  Z)-^-^f  (-•'■)/=^Jcwy+i  •  •  • 

_l_  D'-2^mr.-2mc^^  _|_  Z)-i^f  (— ^)/2]a,„  =  a,-  -  D'^^'^i'\ 

D%'(X2  +  •  •  •  •  •  •  

D%'ai  +  D^o'a2  •  •  •  +  Z)-^-^f  K-.^/2],^  .^^  . . . 

(^  =  1,  2,  3,  •  •  •  to)  and  (j  =  1,  2,  3,  •  •  •  m). 

(7H  —  7  \  TO  "~  1 

— ^ —  1  is  the  integral  part  of  — ^ — ,  and  D  is  the 

Aronhold  operator 

Also  DW^,^[(—i)/2]  =  0  for  j  >  i. 


6  THE   RESOLVENTS   OF  KONIG. 

Solving  for  am-y+i,  we  have 

CXm-j+l  = 

0,  0,  ...  ...,  ai-Z)^f-/% 

0,  0,  ...  ...,  a2-D'^^-"\ 

0,  0,  •  •  •  2)-(i+i)|f  (^-c^+iM],    ai  -  Di^^^"\ 


0,         ^'"-^^o",     ••• 

2)»»-(i-l)fcJ?[(m-y-l))/2]      ,  .  .    £)m-2tE[(m-2)l2-\    J)Tn-l  tE[(m-l)i2] 


-^A. 


where  A  =  ^o*". 

If  we  expand  the  numerator  of  this  expression  in  terms  of  the 
column  tti  -  D'^^'^'^\  the  minors  of  the  constituents  a,-  -  D'^f^'^^ 
are  determinants  of  the  type  (vl)  of  §  1.  So  in  the  symbolic 
notation, 

(2)       Ort^j+i  = 

. . .  y+i^mn*-u+i)my-i^mm-u-i))i2-] 

This  result  enables  us  to  make  a  rapid  computation  of  the  ca  in 
particular  cases. 
Example  1.    m  =  4,  j  =  3. 

a2  =  [{iD\D\D')%mo)%}iai  -  D^o') 


A   QUADRATIC   POLYNOMIAL.  7 

+  D%'D%D%{a^  -  D%')]*  -  ^o^ 
=  [HD^o'D^o  -  D%D%)(a,  -  D^o') 

-  l'^o'DUa2  -  2)V)  +  l-^o^o(«3  -  2)V)]  -  ^0^ 
=  [(^i'  -  ^0^2) (ai  -  2^o^i)  -  ^o^i(a2  -  ^i^  -  ^0^2) 

+  ^o'ias  -  ^1^2)]  4-  ^0^ 
=  [(-  ^0^1^  +  ai^2  -  ^o^ia2  +  ^o'as)  +  2(^o'^i  -  ^o«i)y  ^  ^0^. 
Example  2.    m  =  3,  j  =  3. 
ai  =  [(Z)2(Z)3)V)^^o(ai  -  D^o)  -  (2)H^^)V)^^o)(a2  -  D%) 

=  KD^o-D^o'  -  D%'D%){ai  -  D^o) 

-  D%-D^oKa2  -  D%)  +  Do^,-Do^oKa^  -  D%)]  -  ^0^ 

=  [(^1-0  -  l-^2)(«i  -  ^i)  -  ^o-0(a2  -  ^2) 

+  lo-l(a3- 0)1-^0^ 
=  [^o«3  -  (-  ^1  +  aiM  ^  ^0^ 

Without  interfering  with  the  generality  of  the  results  we  may 
take  ^0=1,  and  then  the  values  for  ai  and  0:2  can  be  put  in  the 
following  form:t 


a,  =  am-      H     K-^^'a^lX 


(3) 
where 


<=0 


\i 


oai  oa2  dm 


Thus,  for  example,  the  results  in  the  particular  case,  m  =  4, 

*  (Z)')*^o'  =  D'~'{o'  =  D°y  is  factored  out  of  each  term  since  it  enters  the 
result  as  a  constant  multipUer. 

t  Cf.  Transactions  Amer.  Math.  Soc,  Vol.  15  (1914),  No.  1,  p.  79. 

t  —  fi  is  substituted  for  x  before  the  differentiation  is  performed  and  (i)  is 
the  power  of  the  differential  coefficient. 


8 

THE   RESOLVENTS   OP   KONIG. 

can  be  written 
(4) 

i=0    U; 

or 


ai  =  tti-  ^2l24'«if,  -  ^2'043ai^y, 


=  04  -  (ao^i^  -  ai^i  +  02)^2  +  ao^2^ 

0:2  =  fi4aifi  +  ^2fi4^0-fi 

=  (—  ao^i^  +  ai^i^  —  a2^i  +  as)  +  (2ao^i  -  ai)^2. 

This  result  agrees  with  that  of  example  1  if  we  make  |o  =  1. 
From  (3) 

^[(w-l)/2]_    tJ+1  ...         ti  ... 

There  is  proved  in  the  paper  just  cited  a  general  result*  which 
is  applied  here  for  the  case  v  =  2;  namely, 

1    f) 

{0i2k+l  +  0;2fc+2a^)   ~    ~  J^qF  (<^2A-1  +  (X2kX). 

Hence  the  assumed  expansions  (1)  and  (la)  take  the  form 
dF  1  d^F 

(6) 

\\ iL  ^LjL_  (ti\kA !^ tl ^ (t  2\E(ml2) 

^    \k     d^^^''^^  \E{ml2)  a|2^c>"^^^^''^ 

The  coefficients  in  this  expansion  are  symmetric  functions  of 
the  roots  of  ^^  and  also  of  those  of  a^.  Expressed  in  terms  of 
the  coefficients  of  a^"*,  and  ^x^,  they  are  linear  in  a^,  ai,  •  -  •,  Om, 
but,  in  general,  not  linear  in  ^0,  ^1,  ^2.  These  functions  have  a 
seminvariant  character  as  appears  in  the  application  which 
follows. 

*  In  Glenn's  paper  the  corresponding  expansions  of  the  general  form  of 
order  i»  in  the  general  argument  ^a,"  is  given. 


A   QUADRATIC   POLYNOMIAL.  \i 

§  3.    CONDITION  THAT  A  FORM  CONTAIN  A  GIVEN 
QUADRATIC  FORM  AS  A  FACTOR. 

Using  the  expansion  (6)  obtained  in  §  2,  a  minimum  set  of 

necessary  and  sufficient  conditions  that  a  given  form  be  divisible 

by  the  rth  power  of  a  quadratic  form  is  the  identical  vanishing 

^  ^   dF    d'F  d'-'F       _.        ^,  .  ,. 

01  F,  -rr ,  wjr-o,  ' '  •   >.^  ,_,  .     Smce  these  expressions  are  linear 

in  X  and  do  not  vanish  identically  for  every  value  of  x,  then  the 
identical  vanishing  of  each  of  these  r  expressions  necessitates 
the  vanishing  of  two  seminvariants,  and  hence  the  simultaneous 
vanishing  of  2r  seminvariants  furnishes  the  minimum  set  of 
necessary  and  sufficient  conditions  that  ax""  be  divisible  by  the 
rth  power  of  ^J^.  In  particular,  if  ax"*  contain  ^J^  as  a  factor, 
then  from  (4)  and  (5) 

j;(m-l/2)  t  i+l 

(7)  '- 

^    '  ^(m-l/2)  t  i 

f=0  [^ 

and  conversely.  These  two  conditions  are  equivalent  to  the 
redundant  set  of  m  +  1  conditions  furnished  by  the  identical 
vanishing  of  the  co variant  $,  given  by  Clebsch,*  which  for 
w  =  4  is 

$  =  -  2D{aO^,'ax'  +  2^x'{a^')ian^Jax  =  0, 

where  D  is  the  discriminant  of  ^^^  and  ^^  =  ^J"^  =  ^x"^-  B.  Igelf 
has  developed  a  method  depending  upon  the  vanishing  of  a  series 
of  resultants  of  every  two  forms  of  a  certain  system. 

*  Theorie  der  Binaren  Formen,  p.  94. 

■\  Sitzungsherichte  der  Kon.  Akad.  der  Wissenshaften  Mathematische.,   Bd. 
LXXXII,  p.  943. 


10  THE   RESOLVENTS   OF   KONIG. 

§4.    SEPARATION   OF   a,-/(y)^   INTO   PARTIAL 
FRACTIONS. 
From  (6)  §  2,  we  have 

dF  1  d'^F 

Hence 

dF^  d'^-'F 

a^  ^  _iL  _     ^^2      ,  (-  1)^^  d|2^ 

{^.'Y      {^.'Y      i^.')'-''^'"    \h-l      ^.^ 


,   (-  1)^    d'F      (-  1)^+^  d^+'F     ..,. 
"^     1^     'd%'^   [Mil  d^s'^+i'^^"^ 

or,  upon  writing  the  numerator  in  explicit  form, 

E[{m-V)li]         y  i+1  y  i 


/2) 


+  E 

To  illustrate,  let 

a/       :x^-  x^-\-Zx^-  X 


i^.')'-'' 


F  dFld^2 


+ 


i^-  x-\-iy      (x^-  x+  1)2  '    \2{x^  -x+1)' 

From  (5) 

F=  -  (^1^  +  ^1  +  3)^2  +  |2^  +  {(-  li^  -  ^i'  -  3^1  -  1) 

+  (2^i+m2}a:, 


A  COMPUTATION   OF  SYMMETKIC   FUNCTIONS.  11 

riF 

~  =  -  (^1^  +  ^1  +  3)  +  2^2  +  (2^1  +  l)x, 

Now,  ^1  =  —  1,  ^2  =  1- 

dF__  _     _         _L^_-, 

Then 

a^-:i^+Sx^-x  x-2  a;  +  1         ,  1 


(a:2_a;+l)3  (x^-x+lf  '   {x'-x+iy  '  x^-x-\-l 


§5.    COMPUTATION  OF  SYMMETRIC  FUNCTIONS. 

The  two  seminvariants  obtained  in  §  3,  whose  identical  vanish- 
ing furnished  the  necessary  and  sufficient  conditions  that  a  form 
of  order  m  be  divisible  by  a  given  quadratic  form,  also  furnish  a 
direct  method  for  the  computation  of  certain  classes  of  symmetric 
functions. 

Let  the  m  roots  of 

fix)  =  ax""  =  aox"^  +  aix"^-^  +  a2X^^  +  •  •  •  +  a^  =  0 

be  Xiy  Xi,  •  • '  Xm,  and  denote  the  totality  of  the  sums  of  these  roots 
taken  in  pairs  by  ^u  and  the  products  in  pairs  by  ^ij-  [i,  j  =  1, 
2,  .  • .  \m{m  -  1).] 

The  number  of  quadratic  forms  which  will  divide  aj^  is 
|w(w  —  1)  and  they  are  of  the  form 

(8)  x^  -  {Xi  +  Xj)x  +  XiXj  =  a;2  +  ^^^x  +  ^2*. 

Hence,  if  from  the  two  conditions  (7), 

i=0  [J^ 

where  ai  is  of  order  m  —  2  in  ^i  and  E  i  — ^ —  J  in  ^2,  and  012 
is  of  order  w  —  1  in  |i  and  E  I  — ^ —  )  '^^  ^2,  we  eliminate  ^2, 


12  THE   RESOLVENTS   OF  KONIG. 

we  obtain  a  resolvent  of  order  |m(w  —  1), 

J»i(m— 1) 

(9)  <!>  =    Z  (-  iy<i>i^i^"'^^-'^-'  =  0, 

whose  roots  are  the  ^i^.  The  0^  are  values  of  the  elementary 
symmetric  functions  of  the  ^u  and  are  rational  functions  of  the 
a's.     By  eliminating  ^i,  we  obtain  in  the  same  manner 

Jm(m— 1) 

(10)  ^  =    E   (-  i)%^2''"^"*-'^-^'  =  0, 

a  resolvent  whose  roots  are  the  ^2y.  The  \f/j  are  the  values  of  the 
elementary  symmetric  functions  of  ^2j  and  are  expressible 
rationally  in  terms  of  the  a's. 

§6.    THE  RESOLVENTS  OF  KONIG. 

The  ^  =  0  of  §  5,  is  the  Konig  resolvent  for  the  case  v  =  2. 
$  =  0  is  a  resolvent  of  a  similar  nature.  Both  of  these  resolvents 
have  been  studied  by  Glenn*  in  his  theory  of  the  rational  resol- 
vents of  the  factorable  ternary  forms. 

The  general  form  of  the  $  resolvent  is,  from  (7) 

12a!^f„  9i'a':fll    •     •    u'^^+y^-t^T'''"^  0,  •    •    •    •    0 


*t  = 


0,        am,     ^'al^„ •" 

0,  0, ^^(.+3/2)^».[^(-l/2)] 


=  0. 


-fl 

The  elimination  of  ^i  from  equations  (7)  does  not  exhibit  the  ^t 
resolvent  in  a  concise  form  like  the  determinant  above. 

The  determination  of  the  ^m{m  —  1)  quadratic  forms  which 
divide  a  given  form  can  be  obtained  by  solving  either  $  =  0 
or  ^  =  0.  If  <E>  =  0  be  solved,  then  by  substituting  the  ^u  in  the 
condition  a2  of  (3),  ^2i  can  be  obtained  and  the  quadratic  factor 
is  determined. 

*  Amer.  Jour,  of  Math.,  Vol.  32  (1910),  p.  80. 

t  The  order  of  the  determinant  is  m  —  1. 

I  The  order  of  the  ^  determinant  ism  —  1+m— 2=2m  —  3. 


THE  RESOLVENTS   OF  KONIG.  13 

These  resolvents  are  given  for  the  case  m  =  4.     From  (4'*), 
ai  =  —  «o^2^  +  («o^i^  -  ai^i  +  02)^2  +  «4  =  0, 

ai  =  (2ao^i-«i)^2+(-ao^i^  +  a^^x  -  02^1  +  as)  =  0. 

Eliminating  ^2, 

—  ao,  ao^i^  —  aili  +  ^2, 

$  =   2ao^i  —  ai,  -  ao^i^  +  axl\  —  o.i^\  +  as, 
0,  2ao^i  —  «!, 

0 

—  ao^i^  +  o.x^\  —  02^1  +  as 
Likewise,  eliminating  ^i, 


=  0. 


^ 


ao^2,  —  ai^2,  —  ao^2^  +  a2^2  +  cii, 

0,  ao^2,  —  ai^2, 

0,  0,  ao^2, 

ao,  ai,  2ao^2  —  a2, 

0,  —  ao,  «!, 


0  0 

—  ao^2'  +  a2^2  +  a^               0 

—  ai^2,  —  ao|2^  +  a2^2  +  a4   =0. 

—  ai^2  +  as,  0 
2ao^2  —  a2,  —  ai^2  +  as 

The  explicit  expanded  forms  of  $  =  0  and  ^  =  0  can  be  ob 
tained  from  the  tables  in  §  7. 


§7.    TABLES  OF  SYMMETRIC  FUNCTIONS  OF  THE 

SUMS  AND  OF  THE  PRODUCTS  OF  THE  ROOTS 

OF  A  GIVEN  FORM,  TAKEN  IN  PAIRS. 

In  §  5,  it  was  shown  that  there  existed  two  resolvents  $  =  0 
and  ^  =  0,  whose  roots  were  the  ^u  and  the  ^2y  of  (8)  respectively, 
and  hence  the  coefficients  of  the  powers  of  ^1  in  $  =  0  and  I2 
in  ^  =  0  are  the  values  of  the  elementary  symmetric  functions 
of  ^u  and  ^2y,  respectively,  expressed  rationally  in  terms  of  the 
a's  of  /  =  fla;"*.    These  symmetric  functions  have  been  computed 


14 


THE  KESOLVENTS   OF  KONIG. 


for  the  cases  m  =  3,  4,  5,  6  and  a  table  of  them  in  homogeneous 
form  completes  this  paper.  In  order  to  check  these  results, 
the  following  formulas  have  been  derived : 


(a) 


(b) 


<f>k=    Z^i«0^'«l'*"-   (im\ 

<-»-(f)=-?^r)-(r)'- -(:)"■ 


As  a  further  check  upon  the  \f/'s,  it  may  be  noticed  that  by  inter- 
changing ao  with  Om,  di  with  ttm-i,  etc.,  \J/j  is  transformed  into 
^^y.     (i  =  0,  1,  2,  .  • .  m.) 


Table  la.^ 


m 


3. 


4>i  —  2aoC5i, 

02  =  fli^  +  ao«2, 

03  =  0,1^2  —  doas. 


Table  lb. 

^,= 

aoa2, 

1^2   = 

aias, 

)/'3  = 

as\ 

Table  Ila.    m  =  4. 


<^l 

<h 

«^3 

<^4 

.^6 

</>6 

Sao^ai 

2a,^a2 
+  3aoai2 

4aoaia2 

-  4ao^a4 
+  aoai«3 

+  2ai2a2 

—  4aoaia4 

+  aia2a3 

Table  lib. 

' 

lAi 

^2 

^Pz 

'/'4 

\^B 

\^6 

ao2a2 

-  ao2a4 
+  aoOias 

-  2a^aia^ 
+  ai2a4 

+  aia3a4 

+  a2a42 

a4« 

TABLES  OF  SYMMETRIC  FUNCTIONS. 


15 


Table  Ilia,    m  =  5. 


<^2 


4>3 


<t>i 


«^5 


4ao'ai 


Sao^a2 
+  QaoW 


dodz 
+  4aoai^ 


+  4ao^aia3 
+  daoW 


—  11  floras 

—  5ao^aia4 
+  2ao^a2az 
+  5ao«i^«3 
+  6aoaia2^ 
+  3ai%2 


Table  Ilia  (continued). 

./>6 

<h 

<^8 

<t>9 

<t>10 

—  22ao2a]a5 

-  ^a^^a^a^ 

loo^aza^ 

400^04^5 

-  aoW 

—  2aQ^a2ai 

—  ^a<^aza^ 

-  4:a,^a^ 

+  4aoaia3a5 

+  2aoaia4«5 

-  a^W 

—  16ao«i^a5 

-  9aoaia2a5 

-  ^aottiGi^ 

+  OoChazas 

—  2a^^a^ 

+  aotta^as 

-  SaoaiazGi 

—  ao«2^a5 

-  aoGz^ai 

+  Gaocnoaas 

+  4aMaz 

+  aoai^a^ 

-  ttoas^ 

-  aiW 

+  aoa2' 

+  aiaz^ 

-  OffiL^az^ 

-  ^aMa^ 

—  aiai^as 

+  2ai%3 

-  ^ai^a^ 

—  aiai^ai 

+  aiazazai 

+  3aiW 

+  aiH2a4, 
+  aiW 
+  1a\a^az 

+  aia^az^ 

Table  Illb. 


1^2 


lAs 


^6 


ao''a2 


—  ao^a4 
+  tto^aiaz 


—  ao^aiGs 

—  2ao^«2«4 
+  ao^az^ 


+  ao^aza^ 

—  3aoaia2«5 
+  aQaiazd^ 


2ao2a52 

—  2aoCiia4a5 

—  2ao«2a3a5 


Table  Illb  {continued). 


«A6 

rf^i 

^8 

^9 

'/'lO 

aoaitts^ 

-  aaa^a^^ 

-  aitts^ 

astta^ 

05^ 

-  Saoazttia^ 

—  2axazai^ 

+  02^4052 

+  aoa4^ 

+  aia^^a^ 

-  aiW 

+  aa^as^ 

+  aia2a^a6 

16 


THE   KESOLVENTS  OF  KONIG. 


■^ 

TtHecTticoeecoeeio^^-Hee 

(NC0t^O5(NQ0(Mi-i00COi-i    e    eOJTtH 

1 1 1 1 1 1 + 1 +++++++ 

■4 

oo(Me(MeeeeTtHcse<s 

t-i-ICOCOCOC^COfNC^OOCOCO 

1 1 1 1 ++++++++ 

-©: 

CS    C3  "o    C    <3      o    53    CS    %H 

cooi  e'=^o^  erHoo  cs 

1 1 1 1 +++++ 

-& 

i^^ifidcl-. 

1— 1   '^    CO   1— 1    I— I    T— 1     © 

1 1 +++++ 

■§ 

e  e  e  ^o  m' 

(N  05  CO  (N  lO 
1    +  +  +  + 

-^ 

III 

+  + 

-§: 

+ 

-§ 

1 

TABLES   OF  SYMMETRIC   FUNCTIONS. 


17 


e       J?  e    >o  <N    e  r? 

^  e     ^    f*   '^lO  eJ^     c 

€"^"e-^-^  €  €  € 

I  +  +  +  I     I     I     I     I     I     I  +  + 


e  CO  (N  e 


e»^>o«»  e  e  e 
e  U  <^  c  e  e 

O5cco5i:oco-*(N  eco 


e    .o  "^^  e  e  e 
.-   „   „ 


"rt  ''^rt  '=^rt  e  G  '^t   *■   " 
e  e  e  "^w  '^w  e 
Tjt  -^  ^  e  e  TtH 


e  e 
e  e 


+  I   +  +  +  I     I  ++  I     I   ++  I     I     I     I  +  + 


■  e 


e  e 


"^c?  II  e  e  e  e  e  „«  ««  .«  c?  °^^ 

(^^0(^^T-lOicocoTt^(Nt^^05co  e  e-*  e  e<N 

I  +  +  +  I  ++   I     I   +  I     I     I   +   I     I   +  +  + 


""   '"    e  e  e  e 


e  Q  e  e  e?  e 


e  e  e  e 


i>coooeeeecocoecseee^^vo 
(Ncoi-iiocoi>TjHTtii-iioi>'*i>-'^  e  e(N 

I  ++  I  ++  I  +  I     I     I   +  I     I  ++  I     I     I     I     I  +  +  + 


<o     -v     ta 

e  e  e    .. 


e  e 


e  e  e 


e  e 


e  e 


js  e  e 


e  e  «  J?  ^  '^     ■     ■ 


J^  cf  e  ^o  «^o  ^o  ^o  %  e  e  'I  1,  1,  1  1>  e  "-.  e  ^^  ^^  e  =1,  ?, 
00-^00  !5e  e  e  csco'*  (Se  e  eeiN  e"^e  e^-^e  « 

i-H00(NCO(NCDQ0Tt<i-iCO<NrJH'^(N(N(M(N    e(Ni»    e(MCO 

+I+I+I+II+II+IIIII ++++ 


e  e 


e  e 


Co       C     C   (N        M     •*    p*     C      «     ■^  e^     ^ 
CCSC3     CS     G     ^eii«j§'(N«       e     Oi-oi 
o  "o    o  '^o  *^o  ,?^    ?3    e     o   e?   «    «    e?    ?, 

TtH(N(:oi-iooeee(Neeeeoei-ieeeee 
iOT-Hco(NT-Hcococot^t^t^iOTjH  ecq^-Hioc^iooeo 

l+ll+llllll+l+ll ++++ 


M      CO      c^      c^  w 

„«  „^  c?  c?  e 


<!■  'S 


e  o  CO 


CO  1-1  i>  t^ 


e.  ^  cT.  e  «„  "^  ^^ 
.    o.    .     -.    '^ 

G      o      o      o    C3     G 

CO  e  e  e  (M  o 


e  e  e  e 


^. 


(M-^<Nio(Noococqco-^coo5  e 


I     I     I     I     I     I     I     I     I   +   I   +   I   +  +  +  + 


18 


THE   RESOLVENTS   OF  KONIG. 


-^ 

*!!?  e            e  e  (M         e        to 

lli"i:fri  11111 

+++ 1 + 1 1 + 1 1 ++ 

■S 

++ 1 ++ 1 1 + 1 + 1 + 

■5 

1 +++ 1 + 1 1 1 ++ 

to       >»                  lO                  to 

1 ++ 1 + 1 + 1 ++ 

+  1  1  1  ++ 

-§: 

^«  If  "^  «^  c? 

1     1   +  + 

-i: 

/I 
1   + 

-5^ 

^ 
> 

TABLES  OF  SYMMETRIC  FUNCTIONS. 


19 


e  if 
I  + 


^  =2.  ^  1. 


''„  « 


+  I 


I  +  + 


to  cs 

(N  e  e  CO  e  e  e 
+  I  +  I  +  I  + 


'-  e  CO 


e   «   e 


«         *ir        a.^         1^      Tf  ^*J         «i         ^         >*^  ^ 

(Nco(NTt<  e(N(M(N  e  e 

I ++ I ++ I  I +  + 


■  to   ?3 


-f  t  €  e  cf 
(N  CO  e  CO  CO 

I + I + I ++ I 


N  to        CO 

e  e  '2. 
e  e  CO 


I 

I  +  + 


THIS  BOOK  IS  DUE  ON  THE  LAST  DATE 
STAMPED  BELOW 


AN  INITIAL  FINE  OF  25  CENTS 

WILL  BE  ASSESSED  FOR  FAILURE  TO  RETTURN 
THIS  BOOK  ON  THE  DATE  DUE.  THE  PENALTY 
WILL  INCREASE  TO  50  CENTS  ON  THE  FOURTH 
DAY  AND  TO  $1.00  ON  THE  SEVENTH  DAY 
OVERDUE. 


OCT  13  1933 

my    12  1933 

lo^rs"   ' 

207an^'5flt'#» 

•  '5  ■.  "■'--. .- 

1 

■'•'JO 

LD21-100m-7,'33 

"--;  tt '-i  TLi  ^  t»  S'«r 


UNIVERSITY  OF  CALIFORNIA  LIBRARY 


